# Full text of "Analytical Mechanics"

## See other formats

```ANGULAR IMPULSE AND ANGULAR MOMENTUM    271
APPLICATION TO SPECL4L PROBLEMS. 212. Ballistic Pendulum.—A ballistic pendulum is a heavy target which is used to determine the velocity of projectiles. The target, which is suspended from a horizontal axis, is given an angular displacement wiien it receives the projectile. Considering the target and the bullet which is projected into it as an isolated system we apply the principles of the conservation of energy and of the conservation of angular momentum. Just before the bullet hits the target the angular momentum of the system about the axis is that due to the velocity of the bullet and equals
T 7, where If is the moment of inertia
of the bullet about the axis, v is its
velocity, and b is its distance from the
axis just before it hits the target, Fig.
123.    The bullet is supposed to hit the
target normally, when the latter is in the
equilibrium position, and to be imbedded
in it. The angular momentum just after
the bullet hits the target is (I +1'} w,
where I is the moment of inertia of the target and co its initial
angular velocity.    Then, by the conservation of the angular
momentum, we have
/'- = (i+1'} w.  /. v = b L+JL ^               (i)
If we suppose the energy lost during the impact to be negligible the kinetic energy of rotation just after the bullet hits the target equals the potential energy of the system at its position of maximum angular displacement. Therefore
| (1+ JO co2 = (M + m) ga (1 - cos a),              (2)
where M and m are the masses of the target and of the bullet, respectively, a is the distance of the center of mass of the system from the axis, and a is the maximum angular dis-
FIG. 123.```